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Income Distribution up to \(\$ 100,000\) The following table shows the distribution of household incomes for a sample of 1,000 households in the United States with incomes up to \(\$ 100,000^{41}\) \begin{tabular}{|c|c|c|c|c|c|} \hline 2000 Income (thousands) & \(\$ 10\) & \(\$ 30\) & \(\$ 50\) & \(\$ 70\) & \(\$ 90\) \\ \hline Households & 270 & 280 & 200 & 150 & 100 \\ \hline \end{tabular} Compute the expected value \(\mu\) and the standard deviation \(\sigma\) of the associated random variable \(X\). If we define a "lower income" family as one whose income is more than one standard deviation below the mean, and a "higher income" family as one whose income is at least one standard deviation above the mean, what is the income gap between higher- and lower-income families in the United States? (Round your answers to the nearest \(\$ 1,000\).)

Step by step solution

01

Calculate the expected value

To calculate the expected value (mean) of the income, we multiply each income bracket with the number of households in that bracket and sum these values. Since we need to calculate the mean, we need to divide this sum by the total number of households (1,000): \(\mu = \frac{270(\$ 10,000) + 280(\$ 30,000) + 200(\$ 50,000) + 150(\$ 70,000) + 100(\$ 90,000)}{1,000}\) \[\mu = \frac{\$ 9,500,000}{1,000}\] \[\mu = \$ 9,500\]

02

Calculate the variance

Now we need to calculate the variance, which is the average of the squared difference of each income from the mean: Variance = \(\frac{\sum{(x_i -\mu)^2 \times p_i}}{1,000}\) where \(x_i\) represent the income brackets and \(p_i\) represent the number of households in each bracket. Variance = \(\frac{(270(\$ 10,000 - \$ 9,500)^2 + 280(\$ 30,000 - \$ 9,500)^2 + 200(\$ 50,000 - \$ 9,500)^2 + 150(\$ 70,000 - \$ 9,500)^2 + 100(\$ 90,000 - \$ 9,500)^2)}{1,000}\)

03

Calculate the standard deviation

The standard deviation \(\sigma\) is the square root of the variance: \[\sigma = \sqrt{Variance}\]

04

Calculate income gap

The income gap between higher- and lower-income families is the difference between the income of families whose income is one standard deviation above the mean and those whose income is one standard deviation below the mean: Income gap = \((\mu + \sigma) - (\mu - \sigma)\) Once you've found the standard deviation from Step 3, use the above formula to calculate the income gap and round your answer to the nearest \$1,000.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Expected Value

In the realm of statistics and probability, the expected value is a key concept often symbolized by \(\mu\). It represents the average outcome you would anticipate from a distribution if you could repeat an experiment an infinite number of times. Think of it as the long-term average or mean value of random events.To obtain the expected value when examining income distribution, as in our exercise, one would multiply each income level by the probability (or in this case, the percentage) of households at that income level. Then, add all these products together. This process provides a single number that gives us a sense of the 'center' of the income data. It could be interpreted as the income a randomly picked household is expected to have if we randomly pick a large number of households for an infinitely long time.

Standard Deviation

The concept of standard deviation, denoted as \(\sigma\), is intimately related to variance and expected value. In essence, it is a measure of the amount of variation or dispersion in a set of values. A low standard deviation means that most numbers are close to the average (expected value), whereas a high standard deviation means that the values are spread out over a wider range.Calculating the standard deviation in our income distribution exercise involves finding the square root of the variance. This step provides valuable insight into the income variability within the household sample. It can tell us, for instance, how widely households’ incomes vary from the mean income we previously calculated.

Variance

Variance is a statistical measure of the dispersion of data points in a data set. More precisely, it captures the average of the squared differences between each data point and the mean. High variance indicates that the numbers in the set are far from the mean and from one another, while low variance indicates the opposite.In the income distribution example provided, the variance is found by taking each household's income, subtracting the mean income, squaring the result, and then averaging those squared differences. This squared step is crucial because it emphasizes larger differences—a feature that makes variance extremely sensitive to outliers in the data.

Income Gap Calculation

The income gap is a critical issue in socio-economic analyses, often referred to when discussing inequality. It is, in simpler terms, the financial distance between different groups of people, such as high-income and low-income families.In our specific example, calculating the income gap involves subtracting the income at one standard deviation below the mean from the income one standard deviation above the mean. This calculation does not merely provide the difference between average incomes of high and low earners but gives us the range within which a larger percentage of the population's income lies based on their dispersion from the mean. This calculated income gap can provide insights into the overall economic diversity and the level of income inequality in the sample.